A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

Abstract

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method.~By applying the discrete energy method and imitating the continuous analysis,~the discrete estimate that resembles the continuous counterpart is obtained proving stability.~We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator.~Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented.~The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier Stokes equations.